to Equations (129). To determine these latter, we substitute Equation 

 (127) into Equation (122) and integrate for the stress function F, retaining 

 only the periodic terms. The total membrane stresses are then given by 



C - P5 J_ F 



"^^■■2h " R2^'ee 



(130) 



''me = - 7- + ^'xx 

 h 



The total normal stresses are obtained by adding algebraically 

 Equations (129) and (130). The greatest normal stresses occur at midbay 

 of the cylinder (x = — ) and where sinnO = ± 1, which corresponds to the 

 trough and crest points of the lobar pattern, respectively. At these 

 points, the twisting moment M^q is zero and thus the normal stresses are 

 principal stresses. The absolute maximum normal stresses occur at the 

 outer shell wall for the trough points. 



Having obtained the maximum principal stresses ^y ^^^ "^/a in terms 

 of the amplitude of out-of-roundness, the geometric parameters of the 

 cylinder, and the applied pressure, the Hencky-Huber-Von Mises criterion 

 of failure discussed earlier, see Equation (34), is employed to determine 

 the pressure at which yielding initiates at the most critically stressed 

 point. Thus, substitution of the m5Lximum principal stresses ^^ and '^p. 



into 



2 2 2 , , 



d =d +d -dd (131) 



y X x@ 



gives an equation relating the initial out-of-roundness, the geometric 

 parameters of the cylinder, the yield point '^.. of the material, and the 



79 



